Scaling of the variance and the quantification of resource monopolization

Behavioral and evolutionary ecologists often attempt to quantify monopolization of mates or food items using indices such as the variance, the coefficient of variation (CV), the coefficient of dispersion (CD), or the opportunity for selection (I). Because of the tendency for the variance to increase with the mean and because of the relationship between variance and the number of competitors, such indices are of limited value for comparisons between systems that differ in the mean number of resources per competitor or in the number of competitors. Here we examine an alternative index of relative monopolization, Q in which the observed variance in resource distribution is expressed as a fraction of the maximum possible variance (assuming discrete resource items) for a given resource abundance, both corrected by an estimate of the variance expected under a random distribution of resources. Q = 1 when the variance in resource distribution is maximum, and Q = 0 when resources are randomly distributed. We demonstrate analytically that (1) on average, Q is independent of mean resource abundance for overdispersed systems and, (2) Q can be used to compare systems with different numbers of competitors as long as the total number of resource units is not larger than the number that can be monopolized by a single individual. We illustrate the advantages of this method using data from studies on feeding competition in fish and on mating competition in crickets.